Each Tengu has a personal name with ... BO 坊 (priest) and also the name of the mountain where he resides.
The mountains of Japan were thought to be the residence of Tengu since ancient times.

修験道の本 ― 神と仏が融合する山界曼荼羅

All Tengu are said to have 翅 wings.
The からすてんぐ Karasu Tengu have a beak like a Garuda bird,
The Tengu from Kuramayama have a 羽団扇 feather fan, a 顔が赤く鼻の高い red face and a long red nose.
The 木葉天狗 / 木の葉天狗 Konoha Tengu wear a Buddhist hood (兜巾 tokin) and band with bells 鈴掛 and a 野袴 rough long pleated Hakama skirt. 野袴
In the Heian period they they were called Amatsu kitsune, Amatukitsune, Tenko 天狐，天狗 / アマツキツネ Heavenly Foxes.
- - - Tenko (Japanese: 天狐) are a type of divine beast in Japanese folklore. After reaching 1,000 years of age and gaining its ninth tail, a kitsune (fox) turns a golden color, becoming a 'Tenko' (天狐 "heavenly fox"/"celestial fox"), the most powerful form of the kitsune, and then ascends to the heavens. They are able to see a thousand ri ahead. They have nine tails unlike the lower-ranked existences, the yako, the kiko, and the kūko. - source : wikipedia -

There is even a skull of such an amatsu oomi kitsune アマツオオミキツネ Honorable Tengu with horns.

There are various spellings in Japanese. Here is another list:
愛宕山太郎坊、妙義山日光坊、比良山次郎坊、常陸筑波法印、鞍馬山僧正坊、英彦山豊前坊、比叡山法性坊、大原住吉剣坊、横川覚海坊、越中立山縄乗坊、富士山陀羅尼坊、天岩船檀特坊、日光山東光坊、奈良大久杉坂坊、羽黒山金光坊、熊野大峰菊丈坊、吉野皆杉小桜坊、天満山三尺坊、那智滝本前鬼坊、厳島三鬼坊、高野山高林坊、白髪山高積坊、新田山佐徳坊、秋葉山三尺坊、鬼界ヶ島伽藍坊、高雄内供奉、板遠山頓鈍坊、飯綱三郎、宰府高桓高森坊、上野妙義坊、長門普明鬼宿坊、肥後阿闍梨、都度沖普賢坊、葛城高天坊、黒眷属金比羅坊、白峰相模坊、日向尾股新蔵坊、高良山筑後坊、医王島光徳坊、象頭山金剛坊、紫尾山利久坊、笠置山大僧正、伯耆大山清光坊、妙高山足立坊、石鎚山法起坊、御嶽山六石坊、如意ヶ岳薬師坊、浅間ヶ岳金平坊、 source : jomon.org/jisho

- quote - Historical Notes.
Tengu mythology was probably introduced to Japan in the 6th or 7th century AD, in conjunction with the arrival of Buddhism from Korea and China. These goblins thereafter appear in Japan’s ancient documents (e.g., from around 720 AD), and are closely associated with Mount Kurama in Japan (near Kibune), the abode of the legendary white-haired Sōjōbō (Sojobo) 僧正坊, King of Tengu.
In Myths and Legends of Japan (1913; by F. Hadland Davis), the Tengu are said to emanate from the primordial Japanese god Susano-o. Tengu lore can be found not just in Buddhist circles, but also among Shinto, Budo, and Ninpo groups. As late as 1860, the Edo Government was posting official notices to the Tengu, asking the goblins to temporarily vacate a certain mountain during a scheduled visit by the Shogun (see Japan and China, by Captain Brinkley). see de Visser’s report.
...
In paintings and woodblock prints, the boar often appears as the steed of the tengu or of their king, Sōjōbō 僧正坊. Sōjōbō is closely linked to famed warrior Minamoto no Yoshitsune 源義経 (1159-1189), one of Japan's most revered samurai. In a well-known legend, Yoshitsune lived among the tengu in his youth and received training in the arts of war from Sōjōbō himself. Note:
The Buddhist martial deity Marishiten is also often shown riding atop a boar.

Another possible interpretation of the above image relates to the following Zen story: “One day a hunter was in the mountains when he happened to see a snake killing a bird. Suddenly a boar appeared and began to devour the snake. The hunter thought he should kill the boar, but changed his mind because he did not want to be a link in such a chain, and cause his own death by the next predator to come along. On his way home he heard a voice call to him from the top of a tree. It was the voice of a tengu. It told him how lucky he was, for had he killed the boar, the tengu would have killed him. The man subsequently moved into a cave and never killed another animal.” Sources: A Field Guide to Demons, Fairies, Fallen Angels, and Other Subversive Spirits (by Carol Mack, Dinah Mack) and Animal Motifs in Asian Art: An Illustrated Guide to Their Meanings and Aesthetics (by Katherine M. Ball). - source : Mark Schumacher -

- quote - tengu 天狗 Lit. celestial dog.
A bird-like goblin frequently encountered in Japanese folk-beliefs, literature and their pictorial depictions. The Japanese demons derive the name from the Chinese mountain god Tiangou 天狗, but also are related to the winged Buddhist deity Garuda. Furthermore, tengu are seen as transformations (keshin 化身) of Shinto deities, yama-no-kami 山の神, mountain guardians often associated with tall trees.
Tengu are of two physical types: karasutengu 烏天狗 identified by a bird's head and beak; and konoha tengu 木の葉天狗 distinguished by a human physique but with wings and a long nose. This type of tengu often carries a feather fan in one hand. Because of its long nose, tengu are associated with the Shinto deity Sarutahiko 猿田彦 who takes on the visage of a monkey, and tengu masks play a prominent role in some religious festivals.

Early Japanese popular tales such as those in the KONJAKU MONOGATARI 今昔物語 (early 12c) portray tengu as enemies of Buddhism, setting fires at temples or tricking priests. Priests who attain special powers through religious discipline, but use these powers for their own ends were thought to enter in the next life the transmigratory realm of tengudou 天狗道.
The earliest representations of tengu are in Kamakura period emaki 絵巻, such as the "Tengu zoushi emaki 天狗草紙絵巻" of 1296 (Nezu 根津 Museum), which criticize arrogant priests who end up becoming tengu.
According to legend, as a boy the famous warrior Minamoto no Yoshitsune 源義経 (1159-89) trained in magical swordsmanship with the tengu king Soujoubou 僧正坊 (Sojobo) near Kuramadera 鞍馬寺 in the mountains north of Kyoto. Tengu frequently are shown in pictures concerning the life of Yoshitsune, including both the Hogen-Heiji 保元平治 battle screens (Metropolitan Museum) and depictions of "Hashi Benkei 橋弁慶" or "Benkei 弁慶 at the Bridge" theme. The Momoyama period daimyo 大名 Kobayakawa Takakage 小早川隆景 (1532-90) supposedly held dialogues with the tengu king Buzenbou 豊前坊 (Buzenbo) on Mt. Hiko 彦.

The character of tengu gradually changed over the centuries.
For instance, tengu were long thought to abduct children, but by the Edo period they often were enlisted to aid in the search for missing children. Similarly, tengu became temple guardians and sculpted images of them were placed on or around temple buildings. Tengu also are associated with yamabushi 山伏 or "mountain ascetics," whose form they often assumed. Tengu often are depicted wearing the yamabushi's distinctive cap and robe. Illustration of tengu increased in popularity and variety during the Edo period, usually reflecting the more positive and even light-hearted conception of the once-ferocious demon. In particular, the long nose of the tengu carried both comic and sexual meaning in ukiyo-e 浮世絵 prints. - source : JAANUS -